Bill and Hector are playing a game at an arcade. Bill starts the next round of the game with 10 points and earns 10 more points each round. Hector starts the next round with 100 points but loses 5 more points each round. Is there a time when they’ll have the same number of points?

Question

Bill and Hector are playing a game at an arcade. Bill starts the next round of the game with 10 points and earns 10 more points each round. Hector starts the next round with 100 points but loses 5 more points each round. Is there a time when they’ll have the same number of points?
The equation below represents the number of points, y, that Hector has after x rounds. Use the equation to complete the table with the number of points Hector has after 4, 6, and 8 rounds. You can solve on paper and then enter your answer on Zearn.
y=100−5x

Rounds (x)	Points (y)
4	
6	
8	
Excellent!
Next, graph the values you just found on the coordinate plane to show the relationship between the number of rounds Hector plays and the number of points he has.
Rounds (x)	Points (y)
4	80
6	70
8	60
You got it!
Next, consider Bill. The equation below represents the number of points, y, that Bill has after x rounds. Use the equation to complete the table with the number of points Bill has after 2, 3, and 5 rounds. You can solve on paper and then enter your answer on Zearn.
y=10+10x

Rounds (x)	Points (y)
2	
3	
5

GPT-4o mini · Answer

To calculate the points that Bill has after several rounds, we can use the equation he has, which is:

\[ y = 10 + 10x \]

Now let's fill in the table for Bill's points after rounds 2, 3, and 5:

For \( x = 2 \): \[ y = 10 + 10(2) = 10 + 20 = 30 \]
For \( x = 3 \): \[ y = 10 + 10(3) = 10 + 30 = 40 \]
For \( x = 5 \): \[ y = 10 + 10(5) = 10 + 50 = 60 \]

Now we can summarize the points Bill has:

| Rounds (x) | Points (y) | |------------|------------| | 2 | 30 | | 3 | 40 | | 5 | 60 |

So, the filled table for Bill is:

For 2 rounds: 30 points
For 3 rounds: 40 points
For 5 rounds: 60 points

Now, to determine if there is a time when Bill and Hector will have the same number of points, we can set their equations equal to each other:

Hector's points: \[ y = 100 - 5x \]

Bill's points: \[ y = 10 + 10x \]

Setting them equal: \[ 100 - 5x = 10 + 10x \]

Now, solve for \( x \): \[ 100 - 10 = 10x + 5x \] \[ 90 = 15x \] \[ x = \frac{90}{15} = 6 \]

So they will have the same number of points after 6 rounds.

At \( x = 6 \): Calculating Hector's points: \[ y = 100 - 5(6) = 100 - 30 = 70 \]

Calculating Bill's points: \[ y = 10 + 10(6) = 10 + 60 = 70 \]

Therefore, both Bill and Hector will have 70 points after 6 rounds.